RealQM has now been extended to a full Schrödinger Equation SE for an atom as an electron density of total charge $-Z$ surrounding a nucleus as a proton density of total charge $+2Z$ surrounding a nucleus kernel as an electron density of total charge $-Z$, as RealAtom + RealNucleus. The total energy includes the kinetic energies of both electrons and protons as well as all Coulomb potential energies including both electron-electron repulsion and proton-proton repulsion. The nucleus here appears as in inverted form of the atom with switched roles of electrons and protons, like a Russian doll system with an electron-proton-electron pattern from nucleus kernel -Z to nucleus +2Z to atom -Z (in basic form).
RealAtom computes binding energies of atoms in eV and RealNucleus binding energies of nuclei in MeV with a change of scale of about $5\times 10^5$ reflecting a change of spatial scale from nucleus to atom, while the change of scale from nucleus kernel to nucleus is much smaller around $10$.
RealQM thus offers a full SE for an atom with nucleus based on a Hamiltonian including all kinetic and Coulomb potential energies. The model is parameter-free modulo the change of scale from nucleus to atom, assuming a mass ratio of 1836 between proton and electron. The computational complexity scales with $Z$.
Note that textbook Standard Quantum Mechanics StdQM including the Standard Model SM does not offer any such complete Schrödinger which is computable. What is offered is (i) SE for an atom with nucleus modeled as a point-wise charge density with zero kinetic and potential energies, and (ii) a shell model of a nucleus consisting of protons and neutrons swimming in a negative potential from a charge density without kinetic and potential energy. StdQM thus does not include the full SE of RealQM.
In short, RealQM offers the first full SE of an atom including nucleus as a unified model in terms of non-overlapping electron and proton charge densities interacting by Coulomb potentials while adding kinetic energies to potential energies to total energy with computations geared to find minima corresponding to ground states.
Preliminary computations show that RealQM can match observations. RealQM may show a way out of the dead-end of uncomputable StdQM of atom including nucleus.
Test case 1: 2H
The basic test for RealNucleus is the 2H nucleus in StdQM viewed to consist of 1 proton and 1 neutron, and in RealNucleus viewed to consist of 2 proton charge densities surrounding a nucleus kernel of 1 electron. This a the nucleus analog of an atom consisting of 2 electrons surrounding a nucleus of 1 proton, that is the $H^-$ ion of the H atom with one extra electron, which is known to be stable. If we then assume that the only electron of the nucleus kernel of 2H has zero kinetic energy and no self repulsion, we get the message that 2H should be stable. RealNucleus confirms by giving a binding $E\approx 1$ MeV including the kinetic energy and zero potential energy from no self repulsion of of the nucleus kernel.
Test case 2: 4He
A more serious test case is the 4He nucleus in StdQM viewed to consist of 2 protons and 2 neutrons, and in RealNucleus viewed to consist of 4 protons surrounding a nucleus kernel of 2 electrons. In this case both kinetic and potential energy of the kernel add to the total energy, and the question is if then the total energy will be negative indicating stability, or not? We use this code realising RealNucleus in a simple implementation with spherical symmetry starting from this input screen with 2 electrons as red spherical nucleus kernel surrounded by 4 protons in a green-blue nucleus:
Pressing start we get the following output showing electron/proton densities in red and total potential in blue as functions of radius in spherical symmetry:
We see that the electron and proton charge densities meet with continuity at the boundary of the nucleus kernel (crest of red curve) with the electron/proton charge density being attracted by the proton/electron charge density into a negative contribution to total energy dominating over kinetic energies, resulting in a total negative binding energy of $E\approx 7$ MeV with a spatial scaling of $4\times 10^5$ between between nucleus kernel and nucleus. We see that the radius of the kernel of the nucleus is about the half of the nucleus.
Notice that a physicist properly trained by StdQM would say that a nucleus kernel of electrons is impossible because electrons are too big to fit and if fitted the kinetic energy would be in the 100s of MeV. But this is not what RealNucleus tells us as displayed in the output figure above: The radius of the kernel is not so small and the electron kinetic energy can remain small because the electron charge density does not have to vanish on the boundary, only meet the proton charge density with continuity.
We understand that the balance of $Z$ electrons vs $2Z$ protons is instrumental to overcome the potential energy from electron-electron repulsion in the kernel of the nucleus. A configuration of $2Z$ electrons combined with $2Z$ protons as a form of neutral kernel as an analog to an atom with $2Z$ electrons surrounding a proton nucleus with the same number of charges, is unstable.
We thus see that RealNucleus explains in particular why a nucleus with an approximately equal number of protons and electrons, can be stable.
PS2 In RealAtom electron densities meet at a free boundary with continuity (and zero normal derivative), and in RealNucleus electron and proton densities meet likewise. This gives an explanation of the fact that electrons and protons do not instantly annihilate under Coulomb attracting, but can coexist by occupying different regions of space meeting a free boundary with continuity of charge density of same or different sign. In StdQM electrons have global overlaying supports which is not compatible with either repulsion or attraction. RealQM resolves this contradiction by assigning charges separate domains in space.